30 research outputs found

    Edge-disjoint rainbow trees in properly coloured complete graphs

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    A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. The study of rainbow decompositions has a long history, going back to the work of Euler on Latin squares. We discuss three problems about decomposing complete graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantine’s Conjecture, and the Kaneko-Kano-Suzuki Conjecture. The main result which we discuss is that in every proper edge-colouring of Kn there are 10−6n edge-disjoint isomorphic spanning rainbow trees. This simultaneously improves the best known bounds on all these conjectures. Using our method it is also possible to show that every properly (n−1)-edge-coloured Kn has n/9 edge-disjoint spanning rainbow trees, giving a further improvement on the Brualdi-Hollingsworth Conjectur

    A generalization of heterochromatic graphs

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    In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose ff-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is ff-chromatic if each color cc appears on at most f(c)f(c) edges. We also present a necessary and sufficient condition for edge-colored graphs to have an ff-chromatic spanning forest with exactly mm components. Moreover, using this criterion, we show that a gg-chromatic graph GG of order nn with ∣E(G)∣>(n−m2)|E(G)|>\binom{n-m}{2} has an ff-chromatic spanning forest with exactly mm (1≤m≤n−11 \le m \le n-1) components if g(c)≤∣E(G)∣n−mf(c)g(c) \le \frac{|E(G)|}{n-m}f(c) for any color cc.Comment: 14 pages, 4 figure

    Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees

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    A 1-factorization F of a complete graph K2n is said to be G-regular, or regular under G, if G is an automorphism group of F acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2 was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2 is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2
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